In order to calculate for every pixel the colorant values corresponding to the appropriate amounts of each colorant to faithfully render the color of that pixel different models have been described in the literature.
It is understood that in the description hereinafter the term colorant amounts can be considered as an equivalent to colorant values as for each output system the value delivered to the system will result in a certain amount of colorant to render the image.
As amounts of colorants can be considered colorants effectively applied to a substrate as known in printing processes both impact and non impact such as offset printing or inkjet printing, but also other equivalents can be understood. One of other possibilities is the use of electrographic printing processes. In photographic materials colorants are generated during development after exposure of the image to the material.
Different State of the Art Models:
Model Using Masking Equations
In “The reproduction of color” by R. W. G. Hunt (1987, Fountain Press England) and in the book “Principles of Color Reproduction” by John A. C. Yule (originally published in 1967 by Wiley & Sons, and reprinted in 2001 by GATFPress), it is explained that a set of 3×3 “masking equations” can be used to relate the CMY colorant densities with the density values in a photographic color original that is to be reproduced. The underlying concept is that the amounts of the individual colorants are adjusted so that the sum of the density of the main absorption of every colorant with the densities of the side-absorptions of the other two colorants matches the original density. Requiring this condition to be true for the RGB densities leads to the 3×3 linear equations that relate uncorrected and corrected RGB densities. The term “masking equations” refers to the original photographic masks that were used to implement this method. The masking equations enable to calculate the amounts of cyan, magenta and yellow colorants. When a color is to be reproduced of which the density is too high to be rendered with just these three colorants, black colorant can be added so that the density of the printed reproduction matches the density in the original. Many variations exist on the above approach.
A first class of variations relates to the way the amount of black colorant is calculated. By calculating first the equivalent neutral density of the desired color, then reducing the amounts cyan, magenta yellow colorants appropriately and replacing this neutral component by an equivalent amount of black colorant, in principle the same color is obtained but with a lower total amount of colorant, since the black colorant replaces three colorants. This technique is called “under color removal” (UCR) or “gray component replacement” (GCR).
A second class of variations relate to the masking equations themselves. Since with 3×3 masking equations only 3 colors can be exactly corrected, higher order terms are usually added to improve their precision. By doing so, exact compensation can be obtained for more colors. The approach that is usually taken, however, is to minimize the root mean square error of the reproduction process over the gamut of printable colors. This is achieved by first printing a large set of color samples, and then using a numerical regression technique to calculate the optimal set of coefficients of the extended masking equations.
Color separation techniques based on the masking equations were a popular approach in the analog color scanners that were developed in the nineteen seventies and eighties, because they could be relatively easily implemented using analog electronic circuitry.
Color separation techniques based on the masking equations suffer from certain limitations. From a theoretical viewpoint, they describe a density modulation based reproduction process. Examples of such processes are the silver-based photographic process or thermal dye sublimation processes. Many print reproduction processes, however, do not work by directly modulating the densities of colorants, but rather by modulating the size of halftone dots (in the case of amplitude modulation halftoning) or the number of fixed sized halftone dots per unit area (in the case of frequency modulation halftoning). The color mixing behavior is rather different for such processes, due to the complex optical and physical interactions between the halftone dots, the light, and the substrate. This explains why the model of the masking equations is not very accurate to control color separation in halftoning based print reproduction processes and that the determination of the optimal coefficients for such processes requires substantial skill and effort.
Model Using Neugebauer Equations
Based on the fact that the halftone dots of 4 different colorants produce only 16 possible distinct combinations of overlap, and on the assumption that the relative position of the halftone dots can be considered random (an assumption that is closely approximated in most real printing conditions), the Neugebauer equations predict the tristimulus values of a color as a function of the effective CMYK dot areas. A detailed explanation of the Neugebauer equations is found in the book “Principles of color Reproduction” by Yule and in the article “Inversion of the Neugebauer Equations” by Marc Mahy and Paul Delabastita in the magazine color Research and Application (published by John Wiley & Sons Vol. 21, nr. 6, December 1996). In the latter article, various interpretations are given to the original Neugebauer equations, as well as a method to improve the accuracy of the model by localizing the coefficients. The localized Neugebauer coefficients are obtained by printing a test chart with the colorant values laid out on a four-dimensional grid, and measuring the resulting colors. By solving sets of equations, the coefficients are determined so that the colors of the test chart are exactly predicted by the Neugebauer equations. Other approaches for improving the accuracy of the Neugebauer equations that are not discussed in the article consist of adding higher order terms to the original equations or increasing the number of color channels from three tristimulus values to a more accurate sampling of the visible spectrum.
Since the Neugebauer equations predict color as a function of colorant values, they need to be inverted in order to solve the color separation problem. The article by Mahy and Delabastita continues by disclosing that—for the three colorant case—this can be achieved by using a three-dimensional Newton-Raphson iterative process or—preferably—by first converting the Neugebauer equations into a 6th degree polynomial of which the six roots are easily determined using robust numerical techniques. Of the six solutions, the one that has a physical interpretation is selected and used to separate the color into the colorants.
The article finishes by pointing out that the method can be extended to the four colorants case, by considering one of the colorants (for example the amount of black colorant) as a parameter of the equations. Effectively, this means that for a given color, the amount of black colorant has to be determined based on some criterion, after which the cyan, magenta and yellow values are determined by inverting the Neugebauer equations. The article does not give hints on how the black colorant parameter should be selected.
In practice, it is not straightforward to come up with a good black colorant generation strategy for the purpose of separating a color into four colorants. In theory it is even impossible to invert a transformation from the four-dimensional colorant space to the three-dimensional color space, since multiple ink combinations yield exactly the same color. Hence, in order to solve this undetermined problem additional constraints are to be imposed.
A first requirement is obviously that for the rendering of an individual pixel with a given printable color, there is a minimum and a maximum amount of black colorant that can be used to print that pixel, and that the selected amount of black ink should fall in between these two extremes. Considerations that help to make the selection comprise: the maximum total amount of colorant that the printing process supports for printing a pixel, robustness of the color balance in the presence variations of the amounts of colorants due to printer instability, visibility of moiré or graininess due to the geometrical interactions between the halftoned separations, metameric robustness if different light sources or viewing conditions are anticipated, etc. . .
The problem becomes even more complicated when additional constraints are imposed on how the colorants are allowed to change along certain trajectories in color space. For example it is considered desirable—if not mandatory—that the amounts of colorant change monotonously along the neutral axis and along the axes from the dark neutral point to the primary (cyan, magenta and yellow) and secondary (red, green and blue) subtractive primaries.
The reason why it is so difficult to fit all the constraints with the four-dimensional Neugebauer equations is because the management of color and colorants are compounded in one single model. No strategies are available to our knowledge that are simple and robust, and enable to separate the color and colorant management problems when using the Neugebauer equations. The complexity becomes even worse when more than four inks are used for printing, for example, when in addition to the standard CMYK colorants also a light cyan and light magenta colorant is used to reduce halftone graininess in the highlight tones.
Models Using Look-Up Tables and Fast Interpolation
Most mathematical models to calculate the amounts of colorants to render a given color require too many computations for the purpose of directly separating all the pixels in an image. For this reason look-up-tables are usually employed in combination with three-dimensional interpolation. The look-up-tables are first populated off-line using the complex separation models. Performing interpolating techniques in combination with these look-up tables enable fast separation of large images. An example of such an interpolation technique is described in U.S. Pat. No. 4,334,240. Other articles that discuss the use of look-up tables and interpolation are “Tetrahedral Interpolation Algorithm Accuracy” by J. M. Kasson, Proc. SPIE 2170, 24 (1994) and “Comparisons of Three-Dimensional Interpolation Techniques by Simulation” by H. R. Kang, Proc. SPIE 2414, 104 (1995).